Setting up a triple integral using cylindrical & spherical coordinates

[calcuseless]

Homework Statement

Inside the sphere x² + y² + z² = R² and between the planes z = $$\frac{R}{2}$$ and z = R. Show in cylindrical and spherical coordinates.

Homework Equations

$$\iiint\limits_Gr\,dz\,dr\,d\theta$$

$$\iiint\limits_G\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta$$

The Attempt at a Solution

$$2\int_0^{2\pi}\int_0^{R\sqrt{\frac{3}{4}}}\int_?^{\sqrt{R^{2}-r^{2}}}r\,dz\,dr\,d\theta$$
Is my upper limit for r correct? How do I find the lower limit for z?

$$\int_0^{2\pi}\int_0^{\frac{\pi}{3}}\int_?^R\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta$$
How do I find the lower limit for rho?

 

[LCKurtz]

Homework Statement

Inside the sphere x² + y² + z² = R² and between the planes z = $$\frac{R}{2}$$ and z = R. Show in cylindrical and spherical coordinates.

Homework Equations

$$\iiint\limits_Gr\,dz\,dr\,d\theta$$

$$\iiint\limits_G\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta$$

The Attempt at a Solution

$$2\int_0^{2\pi}\int_0^{R\sqrt{\frac{3}{4}}}\int_?^{\sqrt{R^{2}-r^{2}}}r\,dz\,dr\,d\theta$$
Is my upper limit for r correct? How do I find the lower limit for z?

Yes the upper limit is OK. Isn't the equation of the plane z = R/2? That would be z on the lower surface.

$$\int_0^{2\pi}\int_0^{\frac{\pi}{3}}\int_?^R\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta$$
How do I find the lower limit for rho?

Again, the equation of that plane is z = R/2. Express that in spherical coordinates and solve it for $\rho$. You will find that $\rho$ depends on $\phi$, but that's OK.

 

[calcuseless]

You will find that $\rho$ depends on $\phi$, but that's OK.

Ah, I didn't understand that concept. Thank you!

$$2\int_0^{2\pi}\int_0^{R\sqrt{\frac{3}{4}}}\int_\frac{R}{2}^{\sqrt{R^{2}-r^{2}}}r\,dz\,dr\,d\theta$$
Is there supposed to be a 2 out in front? I don't know why I put it there and I don't think it is needed for this problem.

 

[LCKurtz]

$$2\int_0^{2\pi}\int_0^{R\sqrt{\frac{3}{4}}}\int_\frac{R}{2}^{\sqrt{R^{2}-r^{2}}}r\,dz\,dr\,d\theta$$
Is there supposed to be a 2 out in front? I don't know why I put it there and I don't think it is needed for this problem.

I don't know why you put it there either. Go with your instincts. If you don't think it should be there, don't put it there.